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We investigate whether a generic multipartite pure state can be the uniqueasymptotic steady state of locality-constrained purely dissipative Markoviandynamics. In the simplest tripartite setting, we show that the problem isequivalent to characterizing the solution space of a set of linear equationsand establish that the set of pure states obeying the above property has eithermeasure zero or measure one, solely depending on the subsystems' dimension. Acomplete analytical characterization is given when the central subsystem is aqubit. In the N-partite case, we provide conditions on the subsystems' size andthe nature of the locality constraint, under which random pure states cannot bequasi-locally stabilized generically. Beside allowing for the possibility toapproximately stabilize entangled pure states that cannot be exact steadystates in settings where stabilizability is generic, our results offer insightsinto the extent to which random pure states may arise as unique ground statesof frustration free parent Hamiltonians. We further argue that, to highprobability, pure quantum states sampled from a t-design enjoy the samestabilizability properties of Haar-random ones as long as suitable dimensionconstraints are obeyed and t is sufficiently large. Lastly, we demonstrate aconnection between the tasks of quasi-local state stabilization and uniquestate reconstruction from local tomographic information, and provide aconstructive procedure for determining a generic N-partite pure state basedonly on knowledge of the support of any two of the reduced density matrices ofabout half the parties, improving over existing results.

Generic pure quantum states as steady states of quasi-local dissipative dynamics